Abstract

It has been pointed out that the two terms effective scattering volume and laser guide star are scientific synonyms. The first term was introduced by the end of 1970 by specialists in atmospheric optics and laser sounding. Information has also been given concerning fluctuations of waves reflected from an object and waves passed twice through atmospheric inhomogeneities. A corresponding mathematical apparatus allows one to estimate correlation characteristics of a laser guide star. The mutual correlation function of random angular displacements of a plane-wave image and the centroid displacements of a focused Gaussian laser beam are considered. The algorithm of optimal correction is introduced in which a priori information is used. Based on this algorithm, the variance of residual jitter of a star image obtained with the use of a laser guide star is evaluated.

© 1998 Optical Society of America

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References

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  1. R. Fugare, “Laser beacon adaptive optics,” Opt. Photon. News.14–19 (June1993).
  2. R. Ragazzoni, “Absolute tip–tilt determination with laser beacons,” Astron. Astrophys. 305, L13–L16 (1996).
  3. V. P. Linnik, “On the possibility of reducing the influence of atmospheric seeing on the image quality of stars,” Opt. Spectrosc. 4, 401–402 (1957).
  4. V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986).
  5. Scintillation, Digest of the International Meeting for Wave Propagation in Random Media, V. I. Tatarskii, A. Ishimaru, Conf. Chairs (University of Washington, Seattle, Wash., 1992).
  6. V. M. Orlov, I. V. Samokhvalov, G. G. Matvienko, M. L. Belov, A. N. Kozhemyakov, The Elements of Theory of Wave Scattering and Optical Ranging (Nauka, Novosibirsk, 1982).
  7. M. A. Kalistratova, A. I. Kon, “Fluctuations of arrival angle of light waves from extended source in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100–1107 (1966).
  8. V. P. Lukin, “Tracking of random angular displacements of optical beams,” in Proceedings of the Fifth All-Union Symposium on Laser Beam Propagation, Part II (Institute of Atmospheric Optics, Tomsk, 1979), pp. 33–36.
  9. V. P. Lukin, “Correction for random angular displacements of optical beams,” Kvantovaya Elektron. (Moscow) 7, 1270–1279 (1980); [Sov. J. Quantum Electron. 10, 727–732 (1980)].
  10. V. I. Klyatskin, Statistical Description of Dynamical Systems with Fluctuating Parameters (Nauka, Moscow, 1975).
  11. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).
  12. M. S. Belen’kii, “Full aperture tilt measurement technique with a laser guide star,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc SPIE2471, 289–296 (1995).
  13. M. S. Belen’kii, “Tilt angular correlation and tilt sensing techniques with a laser guide star,” in Optics in Atmospheric Propagation, Adaptive Systems, and Lidar Technique for Remote Sensing, J. C. Dainty, ed., Proc. SPIE2956, 206–217 (1996).
    [CrossRef]
  14. V. P. Lukin, “Laser beacon and full aperture tilt measurement,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1996), addendum AMB-35, pp. 1–5.
  15. V. P. Lukin, B. V. Fortes, “Comparison of limit efficiencies for various schemes of laser reference star formation,” Atmos. Oceanic Opt. 10, 34–41 (1997).
  16. V. L. Mironov, V. V. Nosov, B. N. Chen, “Correlation of shifting of laser source optical images in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1467–1471 (1982).
  17. R. E. Good, R. R. Beland, E. A. Murphy, J. H. Brown, E. M. Dewan, “Atmospheric models of optical turbulence” in Modeling of the Atmosphere, L. S. Rothman, ed., Proc. SPIE928, 165–186 (1988).
  18. V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).
  19. V. P. Lukin, “Intercomparison of models of the atmospheric turbulence spectrum,” Atmos. Oceanic Opt. 6, 1102–1107 (1993).
  20. V. V. Voitsekhovich, “Outer scale of turbulence: comparison of different models,” J. Opt. Soc. Am. A 12, 1346–1354 (1995).
    [CrossRef]
  21. T. S. McKechnie, “Atmospheric turbulence and the resolution limit of large ground-based telescopes,” J. Opt. Soc. Am. A 9, 1937–1954 (1992).
    [CrossRef]
  22. M. Sarazin, ed., “Site testing for the VLT (very large telescope),” ESO VLT Rep. 50 (European Southern Observatory, Garching, Germany, 1990).
  23. N. Nakato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the European Southern Observatory Conference and Workshop Proceeding, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993) Vol. 48, pp. 521–524.
  24. A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).
  25. J. Borgino, “Estimation of the spatial coherence outer scale relevant to long baseline interferometry and imaging in optical astronomy,” Appl. Opt. 29, 1863–1865 (1990).
    [CrossRef]
  26. V. P. Lukin, E. V. Nosov, B. V. Fortes, “Effective outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 162–171 (1997).
  27. M. A. Gracheva, A. S. Gurvich, “Simple models of turbulence,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 1107–1111 (1980).
  28. R. J. Sasiela, J. H. Shelton, “Transverse spectral filtering and Melin transform technique applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
    [CrossRef]
  29. V. P. Lukin, “Models and measurements of atmospheric turbulence characteristics and their impact on AO design,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 150–152.
  30. R. Ragazoni, ed., “Adaptive optics at the Telescopio Nationale Galileo,” Yearly Status Report (Astronomical Observatory of Padua, Italy, 1996).
  31. V. P. Lukin, “Hybrid scheme of formation of laser reference star,” Atmos. Oceanic Opt. 10, 975–979 (1997).

1997 (3)

V. P. Lukin, B. V. Fortes, “Comparison of limit efficiencies for various schemes of laser reference star formation,” Atmos. Oceanic Opt. 10, 34–41 (1997).

V. P. Lukin, E. V. Nosov, B. V. Fortes, “Effective outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 162–171 (1997).

V. P. Lukin, “Hybrid scheme of formation of laser reference star,” Atmos. Oceanic Opt. 10, 975–979 (1997).

1996 (1)

R. Ragazzoni, “Absolute tip–tilt determination with laser beacons,” Astron. Astrophys. 305, L13–L16 (1996).

1995 (2)

V. V. Voitsekhovich, “Outer scale of turbulence: comparison of different models,” J. Opt. Soc. Am. A 12, 1346–1354 (1995).
[CrossRef]

A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

1993 (3)

V. P. Lukin, “Intercomparison of models of the atmospheric turbulence spectrum,” Atmos. Oceanic Opt. 6, 1102–1107 (1993).

R. Fugare, “Laser beacon adaptive optics,” Opt. Photon. News.14–19 (June1993).

R. J. Sasiela, J. H. Shelton, “Transverse spectral filtering and Melin transform technique applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

1992 (2)

V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).

T. S. McKechnie, “Atmospheric turbulence and the resolution limit of large ground-based telescopes,” J. Opt. Soc. Am. A 9, 1937–1954 (1992).
[CrossRef]

1990 (1)

1982 (1)

V. L. Mironov, V. V. Nosov, B. N. Chen, “Correlation of shifting of laser source optical images in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1467–1471 (1982).

1980 (2)

V. P. Lukin, “Correction for random angular displacements of optical beams,” Kvantovaya Elektron. (Moscow) 7, 1270–1279 (1980); [Sov. J. Quantum Electron. 10, 727–732 (1980)].

M. A. Gracheva, A. S. Gurvich, “Simple models of turbulence,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 1107–1111 (1980).

1966 (1)

M. A. Kalistratova, A. I. Kon, “Fluctuations of arrival angle of light waves from extended source in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100–1107 (1966).

1957 (1)

V. P. Linnik, “On the possibility of reducing the influence of atmospheric seeing on the image quality of stars,” Opt. Spectrosc. 4, 401–402 (1957).

Agabi, A.

A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

Beland, R. R.

R. E. Good, R. R. Beland, E. A. Murphy, J. H. Brown, E. M. Dewan, “Atmospheric models of optical turbulence” in Modeling of the Atmosphere, L. S. Rothman, ed., Proc. SPIE928, 165–186 (1988).

Belen’kii, M. S.

M. S. Belen’kii, “Tilt angular correlation and tilt sensing techniques with a laser guide star,” in Optics in Atmospheric Propagation, Adaptive Systems, and Lidar Technique for Remote Sensing, J. C. Dainty, ed., Proc. SPIE2956, 206–217 (1996).
[CrossRef]

M. S. Belen’kii, “Full aperture tilt measurement technique with a laser guide star,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc SPIE2471, 289–296 (1995).

Belov, M. L.

V. M. Orlov, I. V. Samokhvalov, G. G. Matvienko, M. L. Belov, A. N. Kozhemyakov, The Elements of Theory of Wave Scattering and Optical Ranging (Nauka, Novosibirsk, 1982).

Borgino, J.

A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

J. Borgino, “Estimation of the spatial coherence outer scale relevant to long baseline interferometry and imaging in optical astronomy,” Appl. Opt. 29, 1863–1865 (1990).
[CrossRef]

Brown, J. H.

R. E. Good, R. R. Beland, E. A. Murphy, J. H. Brown, E. M. Dewan, “Atmospheric models of optical turbulence” in Modeling of the Atmosphere, L. S. Rothman, ed., Proc. SPIE928, 165–186 (1988).

Chen, B. N.

V. L. Mironov, V. V. Nosov, B. N. Chen, “Correlation of shifting of laser source optical images in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1467–1471 (1982).

Dewan, E. M.

R. E. Good, R. R. Beland, E. A. Murphy, J. H. Brown, E. M. Dewan, “Atmospheric models of optical turbulence” in Modeling of the Atmosphere, L. S. Rothman, ed., Proc. SPIE928, 165–186 (1988).

Fortes, B. V.

V. P. Lukin, B. V. Fortes, “Comparison of limit efficiencies for various schemes of laser reference star formation,” Atmos. Oceanic Opt. 10, 34–41 (1997).

V. P. Lukin, E. V. Nosov, B. V. Fortes, “Effective outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 162–171 (1997).

Fugare, R.

R. Fugare, “Laser beacon adaptive optics,” Opt. Photon. News.14–19 (June1993).

Good, R. E.

R. E. Good, R. R. Beland, E. A. Murphy, J. H. Brown, E. M. Dewan, “Atmospheric models of optical turbulence” in Modeling of the Atmosphere, L. S. Rothman, ed., Proc. SPIE928, 165–186 (1988).

Gracheva, M. A.

M. A. Gracheva, A. S. Gurvich, “Simple models of turbulence,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 1107–1111 (1980).

Gurvich, A. S.

M. A. Gracheva, A. S. Gurvich, “Simple models of turbulence,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 1107–1111 (1980).

Iye, M.

N. Nakato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the European Southern Observatory Conference and Workshop Proceeding, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993) Vol. 48, pp. 521–524.

Kalistratova, M. A.

M. A. Kalistratova, A. I. Kon, “Fluctuations of arrival angle of light waves from extended source in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100–1107 (1966).

Klyatskin, V. I.

V. I. Klyatskin, Statistical Description of Dynamical Systems with Fluctuating Parameters (Nauka, Moscow, 1975).

Kon, A. I.

M. A. Kalistratova, A. I. Kon, “Fluctuations of arrival angle of light waves from extended source in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100–1107 (1966).

Kozhemyakov, A. N.

V. M. Orlov, I. V. Samokhvalov, G. G. Matvienko, M. L. Belov, A. N. Kozhemyakov, The Elements of Theory of Wave Scattering and Optical Ranging (Nauka, Novosibirsk, 1982).

Linnik, V. P.

V. P. Linnik, “On the possibility of reducing the influence of atmospheric seeing on the image quality of stars,” Opt. Spectrosc. 4, 401–402 (1957).

Lukin, V. P.

V. P. Lukin, “Hybrid scheme of formation of laser reference star,” Atmos. Oceanic Opt. 10, 975–979 (1997).

V. P. Lukin, B. V. Fortes, “Comparison of limit efficiencies for various schemes of laser reference star formation,” Atmos. Oceanic Opt. 10, 34–41 (1997).

V. P. Lukin, E. V. Nosov, B. V. Fortes, “Effective outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 162–171 (1997).

V. P. Lukin, “Intercomparison of models of the atmospheric turbulence spectrum,” Atmos. Oceanic Opt. 6, 1102–1107 (1993).

V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).

V. P. Lukin, “Correction for random angular displacements of optical beams,” Kvantovaya Elektron. (Moscow) 7, 1270–1279 (1980); [Sov. J. Quantum Electron. 10, 727–732 (1980)].

V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986).

V. P. Lukin, “Models and measurements of atmospheric turbulence characteristics and their impact on AO design,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 150–152.

V. P. Lukin, “Tracking of random angular displacements of optical beams,” in Proceedings of the Fifth All-Union Symposium on Laser Beam Propagation, Part II (Institute of Atmospheric Optics, Tomsk, 1979), pp. 33–36.

V. P. Lukin, “Laser beacon and full aperture tilt measurement,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1996), addendum AMB-35, pp. 1–5.

Martin, F.

A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

Matvienko, G. G.

V. M. Orlov, I. V. Samokhvalov, G. G. Matvienko, M. L. Belov, A. N. Kozhemyakov, The Elements of Theory of Wave Scattering and Optical Ranging (Nauka, Novosibirsk, 1982).

McKechnie, T. S.

Mironov, V. L.

V. L. Mironov, V. V. Nosov, B. N. Chen, “Correlation of shifting of laser source optical images in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1467–1471 (1982).

Murphy, E. A.

R. E. Good, R. R. Beland, E. A. Murphy, J. H. Brown, E. M. Dewan, “Atmospheric models of optical turbulence” in Modeling of the Atmosphere, L. S. Rothman, ed., Proc. SPIE928, 165–186 (1988).

Nakato, N.

N. Nakato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the European Southern Observatory Conference and Workshop Proceeding, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993) Vol. 48, pp. 521–524.

Nosov, E. V.

V. P. Lukin, E. V. Nosov, B. V. Fortes, “Effective outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 162–171 (1997).

Nosov, V. V.

V. L. Mironov, V. V. Nosov, B. N. Chen, “Correlation of shifting of laser source optical images in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1467–1471 (1982).

Orlov, V. M.

V. M. Orlov, I. V. Samokhvalov, G. G. Matvienko, M. L. Belov, A. N. Kozhemyakov, The Elements of Theory of Wave Scattering and Optical Ranging (Nauka, Novosibirsk, 1982).

Ragazoni, R.

R. Ragazoni, ed., “Adaptive optics at the Telescopio Nationale Galileo,” Yearly Status Report (Astronomical Observatory of Padua, Italy, 1996).

Ragazzoni, R.

R. Ragazzoni, “Absolute tip–tilt determination with laser beacons,” Astron. Astrophys. 305, L13–L16 (1996).

Samokhvalov, I. V.

V. M. Orlov, I. V. Samokhvalov, G. G. Matvienko, M. L. Belov, A. N. Kozhemyakov, The Elements of Theory of Wave Scattering and Optical Ranging (Nauka, Novosibirsk, 1982).

Sasiela, R. J.

Shelton, J. H.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

Tokovinin, A. V.

A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

Voitsekhovich, V. V.

Yamaguchi, I.

N. Nakato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the European Southern Observatory Conference and Workshop Proceeding, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993) Vol. 48, pp. 521–524.

Ziad, A.

A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

Appl. Opt. (1)

Astron. Astrophys. (1)

R. Ragazzoni, “Absolute tip–tilt determination with laser beacons,” Astron. Astrophys. 305, L13–L16 (1996).

Astron. Astrophys. Suppl. Ser. (1)

A. Agabi, J. Borgino, F. Martin, A. V. Tokovinin, A. Ziad, “G.M.S: a grating scale monitor for atmospheric turbulence measurements. II. First measurements of the wave front outer scale at the O.C.A,” Astron. Astrophys. Suppl. Ser. 109, 557–562 (1995).

Atmos. Oceanic Opt. (5)

V. P. Lukin, E. V. Nosov, B. V. Fortes, “Effective outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 162–171 (1997).

V. P. Lukin, “Hybrid scheme of formation of laser reference star,” Atmos. Oceanic Opt. 10, 975–979 (1997).

V. P. Lukin, B. V. Fortes, “Comparison of limit efficiencies for various schemes of laser reference star formation,” Atmos. Oceanic Opt. 10, 34–41 (1997).

V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).

V. P. Lukin, “Intercomparison of models of the atmospheric turbulence spectrum,” Atmos. Oceanic Opt. 6, 1102–1107 (1993).

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (1)

M. A. Gracheva, A. S. Gurvich, “Simple models of turbulence,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 16, 1107–1111 (1980).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (2)

V. L. Mironov, V. V. Nosov, B. N. Chen, “Correlation of shifting of laser source optical images in the turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 25, 1467–1471 (1982).

M. A. Kalistratova, A. I. Kon, “Fluctuations of arrival angle of light waves from extended source in turbulent atmosphere,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 1100–1107 (1966).

J. Opt. Soc. Am. A (3)

Kvantovaya Elektron. (Moscow) (1)

V. P. Lukin, “Correction for random angular displacements of optical beams,” Kvantovaya Elektron. (Moscow) 7, 1270–1279 (1980); [Sov. J. Quantum Electron. 10, 727–732 (1980)].

Opt. Photon. News. (1)

R. Fugare, “Laser beacon adaptive optics,” Opt. Photon. News.14–19 (June1993).

Opt. Spectrosc. (1)

V. P. Linnik, “On the possibility of reducing the influence of atmospheric seeing on the image quality of stars,” Opt. Spectrosc. 4, 401–402 (1957).

Other (14)

V. P. Lukin, Atmospheric Adaptive Optics (Nauka, Novosibirsk, 1986).

Scintillation, Digest of the International Meeting for Wave Propagation in Random Media, V. I. Tatarskii, A. Ishimaru, Conf. Chairs (University of Washington, Seattle, Wash., 1992).

V. M. Orlov, I. V. Samokhvalov, G. G. Matvienko, M. L. Belov, A. N. Kozhemyakov, The Elements of Theory of Wave Scattering and Optical Ranging (Nauka, Novosibirsk, 1982).

V. I. Klyatskin, Statistical Description of Dynamical Systems with Fluctuating Parameters (Nauka, Moscow, 1975).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

M. S. Belen’kii, “Full aperture tilt measurement technique with a laser guide star,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc SPIE2471, 289–296 (1995).

M. S. Belen’kii, “Tilt angular correlation and tilt sensing techniques with a laser guide star,” in Optics in Atmospheric Propagation, Adaptive Systems, and Lidar Technique for Remote Sensing, J. C. Dainty, ed., Proc. SPIE2956, 206–217 (1996).
[CrossRef]

V. P. Lukin, “Laser beacon and full aperture tilt measurement,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1996), addendum AMB-35, pp. 1–5.

V. P. Lukin, “Tracking of random angular displacements of optical beams,” in Proceedings of the Fifth All-Union Symposium on Laser Beam Propagation, Part II (Institute of Atmospheric Optics, Tomsk, 1979), pp. 33–36.

R. E. Good, R. R. Beland, E. A. Murphy, J. H. Brown, E. M. Dewan, “Atmospheric models of optical turbulence” in Modeling of the Atmosphere, L. S. Rothman, ed., Proc. SPIE928, 165–186 (1988).

M. Sarazin, ed., “Site testing for the VLT (very large telescope),” ESO VLT Rep. 50 (European Southern Observatory, Garching, Germany, 1990).

N. Nakato, M. Iye, I. Yamaguchi, “Atmospheric turbulence of small outer scale,” in Proceedings of the European Southern Observatory Conference and Workshop Proceeding, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1993) Vol. 48, pp. 521–524.

V. P. Lukin, “Models and measurements of atmospheric turbulence characteristics and their impact on AO design,” in Adaptive Optics, Vol. 13 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 150–152.

R. Ragazoni, ed., “Adaptive optics at the Telescopio Nationale Galileo,” Yearly Status Report (Astronomical Observatory of Padua, Italy, 1996).

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Figures (3)

Fig. 1
Fig. 1

LGS formation scheme. NS, natural star.

Fig. 2
Fig. 2

Correlation functions K from Eq. (14) for different sizes of the outer scale of turbulence (parameter c), laser system aperture sizes (parameter b), and beacon altitude X = 10 km.

Fig. 3
Fig. 3

Correlation functions K from Eq. (14) for different sizes of the outer scale of turbulence (parameter c), laser system aperture sizes (parameter b), and beacon altitude X = 100 km.

Tables (1)

Tables Icon

Table 1 Comparison of Efficiencies for Optimal and Traditional Algorithms of Tilt Correction in the Bistatic LGS Scheme

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

ρ im 2 = F 2 x 2   ρ lb 2 + F 2   ϕ F ss 2 ,
ρ lb = 1 P 0 0 X d ξ X - ξ     d 2 RI ξ ,   R R n 1 ξ ,   R , P 0 =   d 2 RI 0 ,   R ,
ρ F = - F k Σ   Σ   ρ S x ,   ρ d 2 ρ ,
K = ρ lb ρ F / ρ lb 2 ρ F 2 1 / 2 ,
I ξ ,   R = a 2 a eff 2 ξ exp - R 2 / a eff 2 ξ ,
a eff 2 ξ = a 2 1 - X f   ξ 2 + Ω - 2 + Ω - 2 1 2   D S 2 a 6 / 5 ,
K = 0 1 d ξ 1 - ξ 0 d κ κ 3 Φ n κ × exp - κ 2 R 0 2 + a Eff 2 4 cos κ 2 x 1 - ξ 2 κ × 0 1 d ξ 1 - ξ 2 0 d κ κ 3 Φ n κ × exp - κ 2 a eff 2 ξ 2 - 1 / 2 0 1 d ξ   0 d κ κ 3 Φ n κ × exp - κ 2 R 0 2 2 cos 2 κ 2 x 1 - ξ 2 κ - 1 / 2 .
Φ n ξ ,   κ = 0.033 C n 2 ξ κ 2 + κ 0 2 - 11 / 6
κ 0 - 1     R 0 ,   a eff ,   x / k 1 / 2 ;   kR 0 2     x ; × Ω - 2 1 2   D s 2 a 6 / 5     1 .
min ρ lb X - α a / 2 R 0 - 1 / 3 ρ F F 2 .
S pl 0 ,   ρ = k   0 d ξ     d 2 n κ ,   x - ξ exp i κ ρ
n 1 ξ ,   ρ =   d 2 n κ ,   x - ξ exp i κ ρ ;
I = I ξ ,   R + ρ 0 1 - ξ / X .
Φ n κ ,   ξ = 0.033 C n 2 ξ κ - 11 / 3 1 - exp - κ 2 / κ 0 2 ,
ϕ lb ρ 0 ϕ F pl = - 2 π 2 0.033 Γ 1 6 2 1 / 3 R 0 - 1 / 3 × 0 X d ξ C n 2 ξ 1 - ξ / X 1 + b 2 1 - ξ / X 2 - 1 / 6 1 F 1 × 1 6 ,   1 ;   - d 2 1 - ξ / X 2 1 + b 2 1 - ξ / X 2 - 1 + b 2 1 - ξ / X 2 + 4 c 2 - 1 / 6 1 F 1 × 1 6 ,   1 ;     - d 2 1 - ξ / X 2 1 + b 2 1 - ξ / X 2 + 4 c 2 .
ϕ lb ρ 0 ϕ F pl = - 2 π 2 0.033 Γ 1 6 2 1 / 3 R 0 - 1 / 3 × 0 x d ξ C n 2 ξ 1 - ξ / X × 1 + b 2 1 - ξ / X 2 - 1 / 6 × 1 F 1 1 6 ,   1 ;   - d 2 1 - ξ / X 2 1 + b 2 1 - ξ / X 2 .
ϕ lb ρ 0 ϕ F pl = - 2 π 2 0.033 Γ 1 6 2 1 / 3 R 0 - 1 / 3 Γ - 1 5 6   d - 1 / 3 × 0 X d ξ C n 2 ξ 1 - ξ / X 2 / 3 .
K d ,   b ,   c ,   X = ϕ lb ρ 0 ϕ F pl ϕ lb ρ 0 2 ϕ F pl 2 1 / 2 .
ϕ F pl 2 = 2 π 2 0.033 Γ 1 6 2 1 / 6 R 0 - 1 / 3 × 0 x d ξ C n 2 ξ 1 - 1 + 4 c 2 - 1 / 6 ,
ϕ lb ρ 0 2 = 2 π 2 0.033 Γ 1 6 2 1 / 6 R 0 - 1 / 3 0 x d ξ C n 2 ξ × b 2 1 - ξ / X 2 - 1 / 6 - b 2 1 - ξ / X 2 + 4 c 2 - 1 / 6 .
ϕ F pl - A ϕ m .
β 2 = ϕ F pl - A ϕ m 2 = ϕ F pl 2 + A 2 ϕ m 2 - 2 A ϕ F pl ϕ m .
β 2 min = ϕ F pl 2 - ϕ F pl ϕ m 2 / ϕ m 2 ,
A = ϕ F pl ϕ m / ϕ m 2 .
β 2 min = ϕ F pl 2 1 - 2 1 / 3 f X ,   b ,   d ,   C n 2 1 + b - 1 / 3 - 2 7 / 6 1 + b 2 - 1 / 6 1 F 1 1 6 ,   1 ;   - d 2 1 + b 2 ,
f X ,   b ,   d ,   C n 2 = 0 X d ξ C n 2 ξ 1 - ξ / X × 1 + 1 - ξ / X 2 - 1 / 6 - 1 + b 2 1 - ξ / X 2 - 1 / 6 × 1 F 1 1 6 ,   1 ;   - d 2 1 - ξ / X 2 1 + b 2 1 - ξ / X 2 2 × 0 X d ξ C n 2 ξ 1 - ξ / X 5 / 3 0 d ξ C n 2 ξ - 1
β 2 / ϕ F pl 2 = 1 + A 2 ϕ m 2 / ϕ F pl 2 - 2 A ϕ F pl ϕ m / ϕ F pl 2
ϕ ss 2 = ϕ F sp 2 a b / R 0 - 1 / 3 × 0 X d ξ C n 2 ξ 1 - ξ / X 2 ξ / X - 1 / 3 0 X d ξ C n 2 ξ 1 - ξ / X 5 / 3 .
ϕ F pl - A ϕ m 2 / ϕ F pl , 2 = 1 - ϕ F pl ϕ ss 2 / ϕ F pl 2 × ϕ m 2 ,
ϕ F pl - A ϕ m 2 / ϕ F pl 2 = 1 - ϕ lb ϕ F pl 2 / ϕ F pl 2 × ϕ m 2 ,
ϕ F pl - A ϕ m 2 / ϕ F pl 2 = 1 - K 2 0 / 1 + ϕ ss 2 ϕ lb 2 ,
ϕ F pl - A ϕ m 2 / ϕ F pl 2 = 1 - 2 b 1 / 3 0 X d ξ C n 2 ξ 1 - ξ / X 3 / 2 ξ / X - 1 / 6 2 0 X d ξ C n 2 ξ 1 - ξ / X 5 / 3 0 d ξ C n 2 ξ .

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